Let R be a commutative ring with unit. We study subrings RX; Y, lambda of RXY = RX-1, ..., XY-1, ..., Y-m, where lambda is a nonnegative real-valued increasing function. These rings RX; Y, lambda are obtained from elements of RXY by bounding their total X-degree above by lambda on their Y-degree. Such rings naturally arise from studying p-adic analytic variation of zeta functions over finite fields. Under certain conditions, Wan and Davis showed that if R is Noetherian, then so is RX; Y, lambda. In this article, we give a necessary and sufficient condition for RX; Y, lambda to be Noetherian when Y has more than one variable and I grows at least as fast as linear. It turns out that the ring RX; Y, lambda is not Noetherian for a quite large class of functions I including functions that were asked about by Wan.
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